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Extending the Prym map to toroidal compactifications of the moduli space of abelian varieties (with an appendix by Mathieu Dutour Sikirić). (English) Zbl 1361.14022

Let \(A_g\) denote the coarse moduli space of principally polarized abelian varieties of dimension \(g\) and \(R_{g+1}\) the coarse moduli space of connected étale double covers of curves of genus \(g+1\). In the article under review, the authors obtain many precise results which pertain to the question of extending the Prym period map \(R_{g+1} \rightarrow A_g\) to a morphism from \(\overline{R}_{g+1}\), the moduli space of admissible double covers of stable curves, to toroidal compactifications of \(A_g\).
As some examples of the authors’ main results, they give a combinatorial characterization of the indeterminacy locus of the Prym map to the perfect and central cone compactifications of \(A_g\). Furthermore, the authors give a geometric characterization of the indeterminacy locus of the Prym map to the perfect cone compactification up to codimension \(6\) in \(\overline{R}_{g+1}\). Finally, the authors give an explicit resolution of the Prym map to the perfect cone compactification up to codimension \(4\).
The authors’ main conceptual point of view in proving these results are Hodge theoretic in nature and fall within the framework of period maps from moduli spaces to classical period domains. In addition, the authors have taken care to provide a nice discussion of both some Hodge theoretic aspects of toroidal compactifications as well as some aspects of the theory of admissible covers. Both of these topics are important to the authors’ proof of their main results.

MSC:

14H40 Jacobians, Prym varieties
14K10 Algebraic moduli of abelian varieties, classification
14H10 Families, moduli of curves (algebraic)
14D07 Variation of Hodge structures (algebro-geometric aspects)
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